Nonlinear Wave Motion

数学科学：非线性波动Abstract: 0070792 ablowitz将继续研究某些具有物理意义的多维非线性波动方程的一类新的解和相关的线性散射问题。这项工作是在首席研究员发现这些解可以与一个整数相关联之后进行的，这个整数被称为电荷。本文将采用解析、计算和近似/渐近技术，得到在光纤阵列研究中出现的一类非线性微分-差分方程的解和性质。最近由首席研究员和合作者对水波进行的联合实验-理论研究表明，经典斯托克斯水波众所周知的调制不稳定性最终可能导致无限维混沌动力学。最近发现了控制这一现象的潜在机制。计划进一步的实验和分析。初步分析表明，这种动态情景也可以在光纤中观察到。大振幅波动系统的动力学通常被称为非线性波动。与理解小振幅现象的实体理论相比，非线性波动的数学描述仍处于早期发展阶段。应用范围从研究大振幅的海浪到光纤中光脉冲的传播。极其稳定的局部非线性波，称为孤子，与这个项目的研究密切相关。由于在光纤电缆上的高数据速率通信中具有潜在的用途，孤子正在世界范围内进行研究。几年前在非线性波领域的数学发现，现在正处于商业的尖端

NSF Award Abstract - DMS-0070792Mathematical Sciences: Nonlinear Wave MotionAbstract0070792 AblowitzInvestigation of a new class of solutions to certain physically significant multidimensional nonlinear wave equations and associated linear scattering problems will continue. This work follows the discovery by the principal investigator that these solutions can be associated with an integer, referred to as the charge. Solutions and properties of a class of nonlinear differential-difference equations which arise in the study of fiber optic arrays will be obtained by employing analytical, computational and approximation/asymptotic techniques. Recent joint experimental-theoretical investigations of water waves by the principal investigator and collaborators has demonstrated that infinite dimensional chaotic dynamics can eventually result from the well known modulational instability of the classical Stokes water wave. The underlying mechanism governing this phenomenon has recently been uncovered. Further experiments and analysis are planned. Preliminary analysis has indicated that this dynamical scenario can also be observed in fiber optics.The dynamics of wave systems with large amplitude is often referred to as nonlinear wave motion. In contrast to the substantial theory available to understand small amplitude phenomena, the mathematical description of nonlinear wave motion is still in an early stage of development. Applications range from the study of large amplitude ocean waves to the propagation of pulses of light in fiber optics. Extremely stable, localized nonlinear waves, called solitons, are closely related to the investigations in this project. Solitons are being investigated worldwide due their potential use in high data rate communications over fiber-optic cables. The mathematical discoveries made in the field of nonlinear waves only a few years ago are now at the cusp of commercial